Dewey
HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
NONLINEAR THREE STAGE LEAST SQUARES POOLING OF
CROSS SECTION AND AVERAGE TIME SERIES DATA
by
Dale W. Jorgenson
and
Thomas M. Stoker
WP #1293-82
**
April 1982
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
NONLINEAR THREE STAGE LEAST SQUARES POOLING OF
CROSS SECTION AND AVERAGE TIME SERIES DATA
by
Dale W. Jorgenson*
and
Thomas M. Stoker*'
WP #1293-82
April 1982
Department of Economics, Harvard University
Cambridge, Massachusetts 02138
Sloan School of Management, Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
NONLINEAR THREE STAGE LEAST SQUARES POOLING OF
CROSS SECTION AND AVERAGE TIME SERIES DATA
by
Dale W. Jorgenson and Thomas M. Stoker
Introduction
1.
The purpose of this paper is to discuss the estimation
of
exact aggregation models by the nonlinear three
stage least squares
method (NL3SLS) of Amemiya (1977), Gallant
(1977), Gallant and Jorgenson
(1979), Jorgenson and Laffont (1974) and others.
In exact aggregation
models there is a unique correspondence between
individual behavior and
aggregate behavior.
This property, known as recoverability
,
makes exact
aggregation models appropriate for the analysis of individual
data,
average data or both in combination.
In this paper we consider estima-
tion using both cross section data on individuals
and average time
series data.
We present the estimator, discuss its properties,
demon-
strate its similarity to linear pooling estimators,
and indicate its
computational advantages over other nonlinear estimators.
The structural form of exact aggregation models for individual
agents is
Y ikt
=
X
kt
where k indexes agents,
q
dependent variables,
6
i
t
x^
(
V
(i = 1,
6)
indexes time periods, y
lKt
=
(x^,
..., x
)
,
i
....
= 1,
q)
..., q are
is a Q vector of pre-
dictor variables which may vary over both agents and time, and
p
is a
vector of variables which vary over time, but are constant across agents
074494*
The Q vector of coefficients
in a given time period.
function of p
for
= 1,
i
&
(p
t>
6),
is a
and an L vector of structural parameters, 8' = (6
...,
Q^)
All restrictions, including cross equation and exclu-
..., q.
sion restrictions are embodied in the form of g^ (p
9),
,
i
= 1,
...,
q.
We write the system of structural equations for each individual
in vector format as:
=
®
(I
*kt
q
^
<V
B
(1)
6)
where:
=
y kt
b
(p
t
e )'
,
(y
(^
=
matrix of order
lkt'
(p
t
,
W'
•'•'
e)\
•
•>
(p
e
q
t
.
e
)')' and
is the identit y
x
q
By averaging (1) over agents for each time period,
q.
we obtain the structural model for averaged data as:
=
yt
where y
(I
q
and x
®
*t }
B
(
V
(2)
e)
are q and Q vectors of averages of y kJ
.
respec-
and x
fct
tively.
The recoverability property is reflected in the fact that (1)
and (2) display the same parameter vector
g
(p
,6).
6
and coefficient vector
Moreover, the forms (1) and (2) are necessary and sufficient
of x
for the recoverability property, if the population distribution
kt
is unrestricted
(c.f.
Lau (1980)).
,
Many examples of exact aggregation models can be found in the
The simplest structural form, which underlies most discus-
literature.
sions of pooling time series and cross section data,
2
is the linear
model.
y
where
(1)
=
ikt
and
8
P
li
t
+
*kt
9
(1 = l '
2i
are vectors of constant parameters.
6
and (2) consists of the elements of 8
,
9 „
i
,
=
••"
q)
The vector
1,
...,
9
of
q,
For illustration of more general exact aggregation structures,
All examples of the
we present some examples from demand analysis.
Gorman Polar Form are exact aggregation models
—
the linear expenditure
system introduced by Stone (1954) and the various S-branch utility models
such as those in Brown and Helen (1972) and Blackorby, Boyce and Russell
In exact aggregation form, each equation of the linear expendi-
(1978).
dure system appears as:
=
y ikt
where
y..
(p
C
it
i
c
P
" bi
i
\
denotes expenditure on good
U>
i
+ b
\t
i
by family k in period t,
denotes total expenditure and p' = (P lt , •••» P
(common) prices at time
...,
c
and b
,
1
...,
b
,
t.
The vector
and x'
9
)
M^
is the vector of
fc
consists of the parameters
c,
= (1, M.,.)-
Of more recent interest are the AIDS system of Deaton and
Muellbauer (1980) and the translog model of Jorgenson, Lau and Stoker
In our format, the AIDS system appears as:
(1982).
=
where
I
-~
v.,
lkt
c.,
I
l
,
i
kt
meters a,,
)
+
t
The vector
\t £nMkt
inHr-
=a o
+
!!
a.
£
+
In p,
it
consists of the para-
j
l
Pn M.
)Mkt
t
in p.^.
a
...,
* n p.it
i.i
are as above, and Jn P
and p
M,
In p n ^
it
£j
rc
+
(a
^ikt
b,
1
,
q
.... h
,
and c,,,
,
...,
il
q
c
,
and
1
x.
kt
qq
=
(M,
kt
,
M,
kt
The translog model of exact aggregation appears as:
.
—
a
y
=
ikt
(
.
+
E b
.
.
in p
.
>
dtf^
Hb.isA R
let
S
where D(p
>
= -1
+
Z
b
D(p
in p.
t
t
t
A
M
\t
u
skt
)
Y lkt .
;
in
mp7 \t
kt -
\
as before, and A
and p
t
fc
gkt>
i
s =
1,
...,
S denotes S demographic variables such as family size, age
of head, etc.
.... b
,
b*
,
..., M
qq
M.
ICC
A
J.KC
The vector
r
.... bj
Kt
A
s
SKl
,
).
6
consists of the parameters a-,
1
m
and b
,
...,
b
Mq
,
and
x^
=
In each of these examples,
(M^,
...,
M^
a
,
b,
,
q
11
in
M^,
the theory of
consumer demand places contraints on the parameters of the model, which
reduces the size of the parameter vector
6
In the next section we formally present the model and stochastic
structure, as well as the assumptions required for estimation.
present and characterize the NL3SLS estimator.
We then
We conclude with an
empirical example involving substitution among different types of energy
in the United States.
Stochastic Structure and Instrumental Variables
2.
We begin by assuming that
ods under consideration, with K
For each
t
= 1,
t
...,
T denotes the time peri-
the size of the population at time
t.
and the averages
we observe the vector p
?y k t
=
(t
In addition, at time
'
N
<
K
:
i.e.
t
o
we observe
t
o
T)
1
set of size
we observe a cross section data
and
y,
kt
for k = 1,
x, t
kt
.
.
.
,
N.
o
o
For individual behavior, we assume that the model (1) is correct up to a stochastic component
section data is then:
e
.
The model representing the cross
The model relatingb y3
for each individual.
v,
kt
y
where u
=
x')
(I
6
(p
t
=
L
Z e.
/
Kl
+
K
I v
_Kl
k
l
,
k
6*) + u
,
K
/
t
(t = 1,
,
t
t
to x
t
and pK
e,
.
l
covariance matrix
ft
e,
.
kt
e.
e,
.
k'
distributed independently of v
4 t and of
t,
k
,
t
,
,
For the moment, we assume that
these
and
v,
:
kt
,
are independent
v,
is distributed with mean
kt
f t'.
t
and
definite covariance matrix
and positive
r
t'
kt
and positive definite
is distributed with mean
or
e,
(Stochastic Structure)
1:
for k 4 k'
(2*)
.... T)
We make the following assumption about the errors
Assumption
is then:
t
for all k'
and
ft
ft
£5
for k
,
1
v
.
v
kt
is
/ k or
t'.
are known;
later we replace
parameters by consistent estimators.
From Assumption
we have immediately
1
that:
E (u u;,)
t
=£
=
and
:
+
(ft
e
,
ft
v
)
t
,
otherwise
,
= t',
(3)
E (e
kt
U
o
P
V
=
%
t
=
t
.
t
o
,
(4)
=
We denote
+ n
P.
by
'
v
e
fl
u
otherwise
,
.
To consider estimation of the complete model, we stack the
equations (1') representing the cross section as
@
Y = (I
where Y'
(y
m
x)
3
y^ o
,
o
,
(p
t
+
9*)
,
e
(5)
,
o
...,
y
m
y
,
o
21t
...
,
,
o
y
2Nt
....
,
y
o
)
o
is the Nq by 1 matrix of dependent variables, X is the N by Q matrix with
kth row x'
Kt
and
,
c
distributed
is the Nq by 1 matrix of disturbances,
O
and covariance matrix
with mean
fi
@)
I
.
We represent the aggregate
equations (2') as
Y
wnere
i
=
11*
is the Tq by 1
x
2
8
1
(p
f
2
,
8*),
(9*) + u
12'
(6)
.
* * * '
IT*
21*
"**•
...,
5j,
6
1
(p
T
,
9*), x^ 6
disturbances, distributed with mean
u
®
*
* *
'
a1
"
*
"
'
2
(
P ' P ' where P = diag (1/
*^i
Pl , 8*),
aT
'
matrix of average dependent variables, f(9)' =
is the Tq x 1 matrix of structural terms,
Q
?t '
(x'
.... x^ 6 (p
q
r
8.(p., 9*),
6*))
and u is the Tq by 1 matrix of
and covariance matrix
1/^).
The model describing all of the data jointly is obtained by
combining (5) and (6), taking into account the covariances (4).
To faci-
litate discussion of the pooling of time series and cross section data,
it is convenient to transform (5)
and (6) to eliminate heteroscedasticity
over time periods and to produce zero correlation between individual and
average disturbances.
Heteroscedasticity is removed by transforming
where P
where
diag
=
'
(I
(I
©
(x)
v
P
—
q
...,
(v^T..,
P"
)
1
Y
)
vIL,)
(10
=
by
@
I
P~
,
standard population size correction, as in:
a
,
(6)
P"
1
)
f
u has covariance matrix Q
(6*) + (I
u
(x)
^-s
P"
1
)
u
(7)
,
I„.
I
The individual-aggregate correlations are removed by a nonsingular trans-
formation of (5) and (7), which is equivalent to replacing
y
v^K
,
and
>^K
^
-'tt'tt
oooo
tt
oo
y*
o
*t
y*
in (7) by
u
-
Q (y t
Q
(
o
=
u*
Q
(
*t
jP-
o
t
x*
y cs )
,
*cs>
'
and Jk
u*
tt'
oo
,
x
on
where
o
IT
t
o
(8)
o
JL
-
Ut
o
v^K
,
>^K
o
o
o
t
ics )
.
o
and e
x
represent
the cross section averages of y, x and e,
,
y
v
cs
cs
-'cs'
and Q satisfies:
Q (n
u
-(n/k
)n
t
o
e
)
q*
= n
(9)
u
where Q can be determined uniquely from consistent estimators of
£2
and
After applying both corrections, we stack the individual model
fi
.
(5)
and
the transformed aggregate model as;
=
(I
X)
s (p
t
,
e*) +
e
,
(10)
y*
=
f* (e*) + u*
,
and denote the full system as;
Y
=
f (6*) + U
(11)
,
and covariance
where U' = (e', u*'), which is distributed with mean
matrix
"
T.
= E(U'U)
©
h
=
"u
©
T
1
The existence of a transformation which separates cross section
and average time series disturbances, such as (8), does not require the
specific stochastic assumptions employed here.
3
However,
on the rplfvance of the correction (8) are in order.
a
few remarks
First, ignoring the
correction (8) in the NL3SLS procedure amounts to ignoring the individualaggregate disturbance covariances, which affects only the efficiency of
the estimator and not its consistency.
Second, the importance of the cor-
rection depends on the size of the cross section relative to population
size.
In many applications,
will
such as the example given below, N/K
o
be extremely small, with the correction leaving the data unaffected.
Typi-
cal numbers for an analysis of U.S. household demand behavior are N = 10,000
and K
70 million in 1972.
Only when the cross section sample size is
10
of the same order of magnitude as the population size will the correction
yield significant benefits; otherwise it can be ignored.
Before presenting the additional assumptions required for esti-
mation of the complete model, we introduce instrumental variables.
often appropriate to treat some of the predictor variables x
It is
and p
as
endogenous for the individual observations, the aggregate observations,
This can occur when the model is a simultaneous equations model
or both.
in exact aggregation form or part of a larger system of simultaneous equations.
For example, in demand analysis the aggregate data on prices can
reflect both supply and demand influences, requiring aggregate instruments,
with individual total expenditure and demographic variables exogenous.
Alternatively, in
study of savings, errors in variables may necessitate
a
instruments for the individual data, while in the average data such errors
may be negligible.
We assume that there are s-vectors
,k=l,
z, _
kt
for instrumenting the individual data and s-vectors
used for instrumenting the average data.
with rows
1
and
z,
kt
t
I
q
Z
z
,
Denote as Z
respectivelv, and as Z the matrix:
z
®
w
Z
o
=
I
q
©
Z
.
..,
N used
o
t
= 1,
...,
T
and Z the matrices
11
Following Gallant (1977), we make the following assumptions:
Assumption
The parameter space of 6, say 0, is compact,
2:
with the true value an interior point.
Assumption
The components of
,
i
= 1,
.
.
,
q,
are twice continuously dif ferentiable in 6.,
j
= 1,
.
..,
L.
0)
and
3:
6
.
(p
9)
,
Fo r the next two assumptions, we use the notation &]
(p
,
.
8.
(p
to refer to the vectors:
3
i
6
(p
i
=
6)
f
(
6
3
il
_
y~6~
'
2
wnere
6.
(P
i
y;
t'
"
(
iQ
'
2
6.
3
P
B
'fTT y
••"
39.
39
8. n
3
n
•••'
'
is tne mth component of 8.
(p
39
n
39.
;
'
9).
Assumption 4A (Cross Section).
The matrix
N
-*
n!
h
K
F
z
^ikt
kt
Ez kt
K
k
N
Z
o
'Z
o
converges to a positive definite matrix as
The Cesaro sums,
°°.
(
—
- x
o
(
o
h
o
x
^kt-
(x
o
kt
kt
(p
B
i
t
6))(
o
(p
£
kt
6
o
i
(p
e
t
o
^kt
J
o
e))
t
•
o
».
~Kt
6
j
e
(p t
•
°
»
.
9)
12
converge almost surely uniformly in
=r.i
:.
1
= 1,
;.
.
.
.
,
_
N :
K
z
suir
~
l
—
..
s'k t
o
for all i,
9,
= 1,
(x
kt
£
£
'
o
(
i
e)
?t
'
£
2
=
s-kt
= 1,
and s'
(x
o
kt
s
o
(
i
pt
B
.
o
»l.
...,q;j,
...,
where
s,
z
£.
= 1,
—
The matrix
+
»,
|iK t
(y
k\***t
|^
i
t
Z'Z converges to a positive definite matrix
The Cesaro sums,
-x;
it
(
(p
>'it-^
(x;
t
s.
4
Cp
t
B
t
9))
,
i
<V
,
e))
converge almost surely in
4=1,
.
.
.
,
L.
The sums
6,
(y
...,
is the s'th component of z
,
Assumption AB (Time Series)
2S I
...,
The sums,
L.
are bounded almost surely for all i = l,
"_
j
jt
6))
-x;B.
(p
t
,
e))
'
,
for all i,
j
= 1,
..., q and
13
^:sc
?s
T
i
s
(x;
,
p
t
t
B))|
.
,
t
izs«p e
i
'
1
u;
s,t
5.
"
(P
t
,
8))L
t
are bounded almost surely for all
= 1,
and s'
L
Assumption
-
N,T
«
-*•
(M*
xo
I
N
is
...,
q j,
1=1,
the s'th component
::
..
z_.
The matrix
5:
—+—
;--
where z w
I,
...,
i=l,
+
>'*)
x'
',
is nonsingular,
where M
xo
and M
Assumption
z
kt
and
(Identification).
6:
is identified by the instrumental variables
(11)
:£
are defined In equation (16) c: the next ;e:
x
z
;
i.e.
the onlv solution of the alnost sure
t'
o
limits
l2a:
»
N K
-* as
O
(i =
T
-*
<=
Y
l
t
v
^t *t
(y it "
5
£
t
i
(p
e))
t'
=
1
:::-
°
(i = 1,
is the true value 9
.
-/
..., q)
14
Assurr-tirr. -
Ass .-'
linear ridels.
1
art
n :-
:;::; = ;-.: standard re
]
and
-1
-A,
5
:
-
:
_s_al
'.--
r:
Jefinitiot
::
i
_
s
titr
rcdel ar
i
-
:r _ _ er:al
A5sur.rr.c-
t
is
ar.alo-
variables for lir.ear
plieitly separated the ass_rrti:h5
tire series rrrel
ir.
se:ere:e HL3SLS estimators for e;:
the
:r:;s settitr. tttdel iff
11a
"
iirsreter
a
of -.on-
- - 1 r
a
:
.irerer.ts
retlaie the usual large sarrle
desitr !=::::: ass_rrti:rs :: lirear thetry.
;:-=
:
111
vector
ir.
:r
:r:;s =e:-
the
:rier t: facilitate the distussicr of
cars set.
the stlution ::
_
A rsra eter
-.
is
ide _ tified in
the almost sure lirit
the true value.
Sinilarlv,
is
necessarily e:_a_
t:
is
identified
tire series If it is uniquely determined
ir.
the
E .'r
.
a
We i:llett all rarareters idertified ir the ;r:ss settitr. ir
.
B
art all if
,
all rarareters
the
identified
renairiri tarareters
ir.
ir
a
the tine series
vector
'-'
ir.
a
vector
r.
3.
rhe BL3SLS e;::r;::r
r
::
(e))'
[i
"
l;
found a;
the
__
V _
al-e ::
-
siiid
miniiaizej
-•
=
a)
e
:
.
-
1
.
::
-.era
©
%
u v>
is a ;::;:;:£-:
explicitly
£:::r£::r
::'
I
as S,
I
5
can
B
writter =:ra
be
;.-
S
=
(8)
S^
+
(6)
S (9),
13
m. th
:
~~
-
a
•
=
-
-:
5
:.
s( 9 ) = (y* - f*
--ere clearly
5"
and
S
section and average nodels
~Lzez z: esrirate
similarly,
5
7
,
f
t
:
~
::"-;;
t*
-
_
:
fox
the
could
idually
I
--
f*
ere NL3SLS objective functions
cross
"-
1>«
for fixed values of the regaining rararete:-
could re niririzei to estiaate
tara-eters could re estimated fros
strair.s
,
(8))'[sy
7
;- ^
;
(c
:
::;:
the estirated values froa cross
B.
data ee:
sectier
ar.d
If 6
* 6 =
minimizing
f
.
then all
1-
tire settee data
::
-
-
-.
-
16
The advantage
sets to be equal, which results in efficiency gains.
froir
pooling lies in obtaining more efficient estimates.
Note that
S
and the moment matrices Z
9
can be evaluated using only
(6)
1
X,
o
and (I
Z'Z
o o
(x)
Z
y~-'
q
'
o
)
Thus for estimating
Y.
or other more restricted parameterizations of B(p
ft
9), only one pass
,
through the cross section data is required to construct these moments.
This is the main computational advantage provided by exact
aggregation models.
The estimation procedure consists of three steps:
consistent estimators of
and
ft
E
ft
U
;
find
First,
second, minimize (12) to obtain 6:
'
third, calculate the asymptotic covariance matrix of 9.
If
9
is
not empty, then we cannot improve upon previous suggestions in the lit-
erature for finding consistent estimators of
and
ft
e
ft
:
u
for example,
Gallant (1977) suggests estimating each equation of the model by
This involves pooling both data sources on a single equation
NL2SLS.
basis
,
forming
ft
as the estimated residual covariance from the cross
section data, and forming
'
ft
u
as the estimated residual covariance from
the average time series data.
The more usual situation is that
a simpler procedure.
9
is empty, which suggests
First, obtain consistent estimates of 6(p
,
9)
by
o
(linear) 2SLS estimation of each equation using the cross section data.
The estimated residual covariance matrix
provides a consistent esti-
ft
E
mator of
ft
even if
9,
is not empty.
Using the consistent estimators
17
of 6(p
G),
,
t
Holding 9°
solve for consistent estimates of 6°, say 6°.
o
fixed at
estimate the remaining parameters of
,
8
by applying NL2SLS to
each equation of the model or NL3SLS to the system as a whole, using only
the time series data.
als,
provides
fi
u
a
The estimated covariance matrix of the NL2SLS residu-
consistent estimator of
usuallv produces eood starting values for
fi
6
u
In addition, this procedure
r
.
'
to use in minimizing
(12).
The objective function (12) can be minimized using a variety °f well
known computational methods.
A convenient one that illustrates pooling cross
section and time series data is the Gauss-Newton process.
To discuss this
method we require the following notation: Let B (e) denote the matrix:
o
<
Pol
B
o
1
>
=
(6)
I
where B
6
> oq
J
is the Q by L matrix with jth column g^
Qi
(p
,
6)
,
i
1,
,
o
j
= 1,
.
.
.
,
*
where
i=
l,
ij^
(0)
Let
L.
ty
(0)
denote the matrix:
*
(6)
*q
<
=
(e)
P
>
is the T by L matrix with t,j element
--..q,
j
= 1,
...,L and
t
= 1,
..., T.
/i<
x'
6^
(p
,
0),
for
q,
18
The Gauss-Newton process is an iterative procedure for finding
9
from an initial value 6,.
9
l
= 9. + A 9. by first linearizing the system (11) with
is updated to 9
respect to
the current value 9.
At the ith iteration,
1
as:
©
Y - (I
X)
8
(p
,
~
9.)
®
(I
X)
pQ
A9.
(9.)
+
E
,
o
(14)
Y* - f* (9.)^
(6.)
i>
Ae
-
+ u
We then apply Zellner and Theil's (1962) linear three stage least squares
method to the model (14), obtaining:
=
A 9.
(M
xo
l
1
+K)
x
(M
+ M
eo
u
)
(15)
,
where
M
xo
M
M
M
eo
x
= B
(9.)'
= B
(9.)'
>o
"o
i
v
e
_1
e
1
(9\)'
(ft"
=
(9V
(n
(Z
o
'Z
o
ooo
_1
X'Z
©
ZCZ'Z)" !') *
(x)
Kz'z)"
_1
(Z
'Z
1
1
!')
B
ro
(9.)
(Y-(I
(x)
q^^
Z' X)
o
)
^
(x)
(fi
l
—' X'Z o
fx)
= ^
ii
_1
_1
(Q
Z')
)
o
(6
i
i
)
(?*-f* (0,))
.
X)
6
(p^
t
,
9.))
i
19
Convergence to
6
is achieved when A 6. becomes sufficiently small.
In
practice, however, we have found Hartley's (1961) suggestion useful,
namely, we check whether S(9. + A 6.)< S(9.)
by forming
ment in
1
at which point a new iteration is performed, or,
is found,
S
we shrink
if not,
r_.
Checking continues until either improve-
+ A 6./2.
= 9.
9
;
if
the current increment falls below a convergence criterion,
9=6..
we have found
l
Under our assumptions the NL3SLS estimator
T *
9* as N,
<=,
5
is consistent for
and asymptotically normal with asymptotic covariance
matrix:
AVAR
=
(6)
(M*
xo
+ M*)" 1
x
(16)
,
where the *'s indicate (15) evaluated with the true values H
(c.f.
Gallant (1977)).
and
.".
The precise form of the limiting normal distri-
bution depends on the way that N and T approach infinity; however, a
+ M )~
consistent estimator of AVAR (6) is (M
in any case, so this
problem is of secondary concern.
Several points of interest arise from a closer inspection of
(15).
if 6° =
First, the relation to linear pooling estimators is apparent; for example,
9
= 9,
then:
A 6.
= (M
xo
l
where A
6.
l
)~ x (M
+ r.
M v-1
x'
and A 9. are the
l
xo
A 6"
l
+ M
Gauss-Newton
x
A 6.)
1
,
increments from minimizing
::
=
.
.-
:
;
=
'-
i
I - . I '
-
.
i
'IT
:.~r
V
-
-_-
---
-
---------
------
--z
.
-
'
-
\
.
-
,_--:
-
OTU 0* -
-
---------
-
--------
-
'),
- i
I'
Z*
U
'-
u
:.:::-
r-=
ilch satisfies g(e
=
t
1
::=::::.:£: ::r i_.
.
_-
tions
;
r
.:
-
(p)
= (Y - *(g(c
and
;;::£-;-
=
-
r
1
:
?
•-- .- ---..--
og
—Ls.
.
'
- -
-
-
-
-
-
-
:
"
;
--:er
:'- =
-.'-'.
-
-
-
::
'
5_
.
r-it£ :£5t statistic
.
-
:;:-.:;
:
Gallant
::~
)CL
Ls
:
:;
:
5
|
£ ~
f.
.1
of S
sis
to
:
-T£;
aE
is
r
-1
pro
tiod
idek
:
is
-
r.
analogous
the
to
~;-i~~ as t r i c e = :r;~
r.
~ =
;:
22
of
S
Z
can be used in evaluating (17), the desirable monotonicity condition
(p) - S(5)
both
S
r
will be guaranteed if the same
>
and S.
is used to evaluate
I
the original consistent estimates
Thus,
and
ft
e
in estimating
9
ft
u
used
should be used in finding estimators for restricted ver-
sions of the model.
The final topic we consider is the estimation of 9* subject to
inequality restrictions.
For example, in demand analysis, an integrable
demand system must obey the condition that the Slutsky matrix of compensated price derivatives is nonpositive definite.
9
The unconstrained
estimator
need not obey such restrictions for finite samples: thus, it
may be desirable to impose them.
We represent such restrictions for-
mally as
*
m
(6) >
—
where we assume
ponent of
<£
m
m
,
= 1,
.... M'
(19)
,
dif ferentiable in each comto be twice continuously
J
9.
The inequality constrained estimator
the constraints
(19).
9
minimizes S(9) subject to
This estimator corresponds to a saddlepoint of the
Laeraneian function:
f, =
where
X
S(9) +
A'<J>
is a vector of M'
constraint functions.
(20)
,
Lagrange multipliers and
e
£
=
v
is the M' vector of
The Kuhn- Tucker (1951) conditions for a saddle-
point of this Lagrangian are:
V
$
S(9) +
V
($(9))
=
,
23
and the complementary slackness condition:
=
X'<$>
X
,
>_
,
-
r
where *(8) is the
M'
byL matrix with
i,
:
.
.
1
;
To obtain the estimator
in (14).
we begin by linearizing the model as
E
Next, we linearize the constraints as
c(e.
where
.
-
.~
element
j
+1
)
=
i
(e.)
i
+
a
i
(5.)
i
,
is the current iteration value of the unknown Darameters.
6
He
i
then apply Liew's (1976) inequality constrained
linear three s:age lea;
squares method to the linear model, obtaining:
=
a"e
i
&Vl
+
Cm
x:
_1
+ m
x
x*
(e.)'
i
)
i
,
where & 0. is giver, by (15) and \* is the solution of the linear complc
mentarity problem
i
(5.)
(M
l
x:
+ M
_1
x)
i
(9.)* X
l
+
f
& 6
B
-
-
(9
i
)
-
>
—
,
where
i
(e\)
1
[m
XO
* S
r1
X'
i
5.)
1
f
x
+
[i
a
x
&e
]
X
>
x
t
(e )]'
x
•
=
:.
24
Given
+
6.
A
9.
i
i
m
that satisfies the constraints (19), we update to
6
=
1,
and check that both S(e...)
l+l
..., M'.
<
S
(6,), and that
l
<j>
6
=
>
m (6.,,)
i+i —
0,'
If not, we shrink the increment vector as before, until
either improvement is found or the increment values fall in absolute
value below
a
convergence criterion.
the NL3SLS estimator.
This concludes our discussion of
25
Example:
4.
Household Energy Consumption
The purpose of the model we employ to illustrate the nonlinear
three stage least squares estimator is to characterize household demand
The basic assumption of this model is that each household
for energy.
First, total expenditures are
performs a two stage budgeting process:
allocated among all other goods and an energy aggregate; second, total
expenditures on energy are allocated among four individual components;
electricity, natural gas, gasoline and all other fuels.
Since the rela-
tive prices of the components under consideration are not constant over
time,
the existence of an energy price aggregate for each household
requires homothetic separability of the four types of energy and other
,
goods.
9
As is now standard in demand analysis, we begin with the indirect (dual) subutility function corresponding to the four types of energy:
P/
Pi
V
V
——
f-=^
(21)
)
denotes total expenditures on energy for family k in year
where M
the p.
=
,
i
= 1,
...,
A,
t
and
denote the prices of the four energy components.
We assume that all relevant differences among households can be parametrized
by a vector of characterisitcs A^
,
so that
(21)
can be rewritten as:
26
V
=
(—,...
V
kc
)T
^
where
V
types
::"
as
c:~«
is
,
a.
)
(22)
,
Kt
.-•-
to all househrlcs.
The relative demands for all four
energy are found by an abdication of Rcy's (1943) Identity
:
••
:_
a
.
** v
.--.
ikt
P^/H^
In
:
I
3ln
1
Pqt
/\ t
(i = 1,
•--.are
ponent,
i =
A)
,
(23)
the share of energv expenditure allocated to the ith con-
15
-"_...
...,
1,
.
.
.
4.
,
To make the above model operational, we postulate a translog
form for the function V:
p.
4
=
it
a,
Z
+
L- rr^-
4
=
p
4
X
J
b,
"«
i=l j=l
let
,
in
""
—
- it
"kt
p.
r it
*n
""
"kt
(24)
+
_
b
,
1=1 s=_
.
is
As.-;t
.
I
= -1
a,
:
,
i=l
:
i=i
b
A
1S
=
o
=
b
—
tz
M. „
let
The required homogeneity of degree -1 of
i=l
in
.
,
*.
irt:.ies
;
= 1, ..-,
4,
1J
(25)
s
= l,
...,
s.
27
The system of demand functions (23) can be generated by atilit
maximization under symmetry, that Is, b.
—
monotonicity
« b.
.
i,j = 1,
.,
Symmetry can be imposed directly, while mono-
tonicity restrictions take the form of inequalities:
factorized as B = LDL'
...,
4,
...,
,
nst
for all positive share values the matrix B = (b..)
be nonnegative definite.
D = diag (5
...,£, and
where L is a unit lower triangular matrix
,
B is nonnegative definite if 6.
).
£
Let B be Cholesky
where from (25), at least one
'£
is zero.
.
0,
i
£
= 1,
i-pose all of the
"<.£
homogeneity and symmetry restrictions at the outset and impose the monotonicity restrictions if they are required.
Application of Rcy's Identity (23) to the translog form (24)
yields share equations of the forn
"
V
ikt
= a
XV
+
±
+
In P
3t
"is
s ;,
A
skt
.
these are the structural equations of the model; we add
term
E *.
.
lkt
=.
-.
** (26)
stocbastic
:
4
-w.,
lkt
= a.
l
+
S
+
£n o.
it
J
b..
I
.
liJ
.
j=l
=
We assume that the vector e*
kt
zero and covariance matrix
7i*
£
holds.
(i - 1.
Ve treat
M.
and
A,
,
s
b".
I
,
s=l
is
A
.
skt
+
s*.
lkt
.
(27)
is distributed normallv with ~ea~
(e*
)
lkt
and uncorrelated over time and over house-
as exogenous;
therefore,
28
with E*
we assume that these variables are uncorrected
t
.
Notice that
of the indivican be interpreted as representing omitted attributes
e*
kt
included attributes
dual household which are uncorrected with the
A^.
squares estimator
We can apply the nonlinear three stage least
of the model
to the individual and aggregate versions
(27), noting two
required for this
differences between our formal development and that
can be replaced
First, average budget shares in the population
example.
aggregation of (27), using
by market budget shares, defined by weighted
total energy expenditures:
^
ZM^
w
7 M.
W
w
_
,
.
it"
S
uA
+
4
ikt =
a
+
x
jV
^
isT
s=l
.
A
b.
xi
I
.
j=1
skt
Z
,
(28)
la p, t
It
\t (£ ikt
EM*t
+ v lkt >
?*
as before.
where we have added the stochastic term
kt
J
v
kt
=
\
l
v
The vector
with mean zero and
is assumed to be normally distributed
*
1
ikt J
covariance matrix
fl*,
and uncorrelated over households.
the error correcThe aggregate form (28) necessitates altering
tion transformations (7) and (8) of Section
2.
It is easily seen that
-homoscedastic if each equation at
the aggregate share errors will be
time
t
is multiplied by /iT (previously S%~)
(I
.
t
~
V
2
,
where
2
(29)
29
Removing the individual-aggregate error correlations is a more difficult
Unfortunately, the correct transformation (previously (8))
problem.
utilizes the precise distribution of
K,
in each cross section data base.
An alternative procedure, which sacrifices a slight amount of efficiency,
is to use as aggregate data the market shares of all individuals not
observed in the cross section in each year.
However, as indicated earlier,
our cross section observations represent such a minute proportion of the
total population that the aggregate data would be unaffected by removing
Consequently, we ignore the individual-aggregate corre-
them.
lations.
The second difference between the models (27) and (28) and (1')
and (2') arises from the fact that both the individual budget shares
w.,
and market shares w.
that
e
lkt
add to unity, i = 1,
it
kt
and v
matrices.
kt
.... 4.
have singular distributions, with Q e and
This implies
singular
fi
v
We remove this problem by omitting one equation from the model
for estimation (setting q = 3)
;
we can solve for estimates of the para-
meters of the omitted equation from the estimates for the first three
equations.
12
'
The error vectors of the system of three equations are
assumed to satisfy
Assumption
We are now in
in detail.
a
1.
position to discuss the empirical application
As indicated above, our objective is to analyze the alloca-
tion of total expenditure on energy among four types of energy.
The basic
consumer units are households; we differentiate households amons five
demographic characteristics
and type of residence.
—
family size, age of head, region, race,
30
We have represented these characteristics in the attribute vector A^ as qualitative or dummy variables.
represented by family size
White and urban residence.
1,
The base case household is
age of head 15-24, region Northeast, race
This type of household has shares given by
the constant terras.
For the estimation of the model, we utilize five cross section
data sets:
the 1960-61, 1972 and 1973 Consumer Expenditure Surveys (CES)
of the Bureau of Labor Statistics, and the 1973 and 1975 Lifestyle and
Energy Use surveys performed by the Washington Center for Metropolitan
Studies (WCMS).
Data on the four energy categories are quite compatible
across these surveys.
13
"
There are 13,098 observations in the 1960-61
CES, 8879 in the 1972 CES, 8898 in the 1973 CES,
579 in the 1973 WCMS
and 2970 in the 1975 WCMS, making a total of 34,424 cross section obser-
vations.
The aggregate price and quantity series are obtained annually
from the National Income and Product Accounts, for the years 1946 to
1978.
The quantity series are taken as the corresponding components of
personal consumption expenditures in constant dollar form, with the price
series the associated implicit deflators.
The statistics IMA/EM in (28)
are shares of energy expenditures for each demographic group.
These are
constructed on an annual basis following the technique of Stoker (1979)
which utilizes income distribution data obtained from the Current Population Reports, Series P-60.
31
Since we view the demographic variables as exogenous, we
require no instruments for the cross section data (setting Z
= X respec-
tively for each cross section), and utilize OLS on each equation for
getting starting values and consistent estimates of
14
fi
.
In addition,
in order to compensate for slight differences among data sets, we utilize
only within-sample
deviation from means
moments and pool the various
data sets, weighting by the within sample covariance matrix estimates,
again sacrificing a slight amount of efficiency.
The aggregate price data are viewed as endogenous, and so we
utilize fourteen additional variables as well as the attribute-expenditure statistics as instruments.
consistent estimates of
0.
u
We obtain both starting values and
via the technique described above
—
a
estimat-
ing the demographic coefficients with cross section data and while holding them constant, estimating the price coefficients with the time
series data.
Results of the pooled estimation are presented in Table
1.
The
estimates were obtained by fitting the equations for electricity, natural
gas and other fuels;
estimates for the gasoline equation derived from
homogeneity and symmetry restrictions.
We see that most price coefficients
are estimated precisely, except for the coefficients of the price of gaso-
line and the own-price coefficients of natural gas.
This leads us to
test the hypothesis that gasoline and the other fuels are separable.
Also, it is easily seen that the inequality restrictions required for
monotonicity hold, so they need not be imposed.
:-:
Notation
1:
IABLE
-=re-de-: Variables
.
.
,
lze
f
5:
V
....-
f
rr lighting,
«3for
sraft'ssar—
-
—niS^«
"
W3AS:
electricity
::~:;
"
;
water, and
heating, cooking, heating
—^ «
-
._
::
plu5 all
.
la-o -overs
aeration of autonobiles,
and so on.
vehicles,
--"V
heating, use of aopliance
Predictor Variables
P
trice of electricity
,
price of natural gas
.
E
and others,
price of fuel oil
-
P
SOL
price of gasoline
constant (a^
.
,
'
F4, 73, F6, 77
- J
F2
rz » F3,
'
/^-n,
e
-
or riore
'
-:, A70 ,30, A40, A50,
..„
m
6,
« 2 e effects
«ft«c for sizes 2, 3, 4, 5,
s
size
.
£« w
f
25-34, 35-44,
ts for age classes
effects
headrffec
^^
rtl
=en«al
-
region effects fcr
,
Bonvhites,
race effect for
.
rural residence
residence effect for
weighted
f ur.
cci o
r.
5
—
.,
s
squared residuals:
.
-1-
=f
the
^"^
;
:
TABLE
1
Five Cross Sections*
.
r
GAS
'SOL
PGASO
c
.
m
WE
-.
12551 (.0150)
.16616 (.0300)
-.05383 (.0182)
-.05383 (.0182)
.02768 (.0189)
.04653 (.0101)
-.12651 (.0150)
.04653 (.0101)
.10882 (.0188)
.01418 (.0268)
-.02038 (.0178)
-.02884 (.0136)
-.19601 (.0063)
-.05782 (.0055)
-.17469 (.0051)
.00579 (.0026)
F2
.00876 (.0038)
.01163 (.0034
F3
.0169S (.0045)
.00605 (.0041]
-.01072 (.0031)
F4
.00510 (.0049)
-.00069 (.0044]
-.00272 (.0033)
F5
.00114 (.0058)
-.00940 (.0052)
-.01298 (.0039)
F6
.02739 (.0072)
.00184 (.0065)
-.02756 (.0049)
n
.01866 (.0076)
.01868 (.0068)
-.03521 (.0051)
-.04522 (.0056)
-.03043 (.0051
-.:.-'':
A30
A40
-.03068 (.0061)
-.05099 (.0054]
-.02067 (.0041)
A50
-.03498 (.0057)
-.0435S (.0051)
-.03410 (.0039)
A60
-.02949 (.0057)
-.05042 (.0051)
-.05426 (.0039)
A70
-.06150 (.0056)
-.:£:":
-.10285 (.0038]
EJC
.00845 (.0039)
-.07266 (.0035)
.09626 (.2026)
BS
-.04511 (.0039)
-.03367 (.0035)
.11663 (.0026)
RV
.00627 (.0041)
-.05039 (.0037)
.10528 (.0027)
BLK
-.01643 (.0045)
-.08812 (.0040)
.00493 (.0038)
.01289 (.0036)
.11150 (.0032)
-.11125 (.0024)
EL'S.
.::::
Asymptotic standard errors ir. parenthesis
Convergence After 3 Iterations
*
--
:::s
*
.
- -
JAT
.01418
34
Turning to the demographic coefficients, we see that all are
significant at the one percent level except for family sizes
4,
2,
5,
family
North Central and West regions in the electricity equation,
sizes
3
through
7
or more in the natural gas equation,
family sizes
2
and
7
sizes 4,
and 4 and Nonwhite the fuel oil equation, and family
more in the gasoline equation.
6
or
The lack of significance of these coeffi-
indicates no differcients is not a problem for our model, since this
the base case family
ence between households of each of these types and
of size one,
residence.
and urban
age of head 15-24, region Northeast, race White
standard
Moreover, for all the demographic coefficients the
coefficient actually
errors are quite small, so that an insignificant
represents a precisely estimated small effect.
Table
The price effects implied by the estimates in
easily interpreted in elasticity form.
elasticities for each type of household.
1
are most
There are different sets of price
A complete display of all own
vector would require comand cross price elasticities for a given price
types of families.
putation of elasticities for each of the 672 different
in Tables
Since this would be very cumbersome, we present
2
A-E own-price
demographic characelasticities for all four energy types, varying each
teristic while holding prices at 1972 values.
In interpreting the own-
represent elasticities
price elasticities one must bear in mind that they
expenditure constant.
of the energy types holding total energy
For example,
on a given family's conthe effect of an increase in the price of energy
our elasticities, which
sumption will only be partially represented by
35
TABLE
2
Own Price Elasticities
A.
Varying Family Size
3
2
7
A
oi
llectricity
-1.732991 -1.762442 -1.7Q2333 -1.740P47 -1. 736699 -1.833738 -I.
latural Gas
-1.254490 -1.284875 -1.269492 -1.252834 -1.234197 -1.7SP799 -1.
"uel Oil &
-1.557023 -1.574029 -1.52804* -1.549385 -1.522322 -1.4««157
Other
-1.074701
lasoline
-1.0707S4 -1.0727*7 -1.074433 -1.071243 -1.074437 -1.
B.
25-34
15-24
electricity
latural Gas
?uel Oil &
Other
Jasoline
Varying Age of Head
45-54
35-44
55-64
-1.419793 -1.374565
-1.579846 -1.537328 -1.522322 -1.490*83
-1.084232 -1.116745
-1.063697 -1.076230 -1.078243 -1.080090
N.E.
N.C.
Varying Region
S
W
Electricity
-1.736*99 -1.765390 -1.613920 -1.757778
Natural Gas
-1.234197 -1.145039 -1.182277 -1.164196
-1.522322 -1.970945 -2.186580 -2.055859
Fuel Oil & Other
-1.078743 -1.073018 -1.072146 -1.068844
Gasoline
Varying Race
Nonwhite
White
D.
Varying Type of Residence
Rural
Urban
-1.736699
-1.686668
-1.736699
-1.781366
Electricity
-1.234197
-1.134174
-1.234197
-5.125896
Natural Gas
-1.522322
-1.534976
-1.522322
-1.340497
-1.078243
-1.100625
-1.078243
-1.076013
Gasoline
65 or more
-1.740575 -1.64PU4
-1.8526*7 -1.692060 -1.736699 -1.722308
-1.249*60 -1.235326 -1.181041
-1. 411*47 -1.283502 -1.234197
C.
Fuel Oil
&
-1
Other
36
reflect substitution among energy types, but not substitution between
energy and other goods.
In general our results indicate that families
are very sensitive to energy price changes, given the level of total
energy expenditures.
To illustrate tests of hypotheses about household demand for
energy using the model (27) and (28), we test the hypothesis that gasoline is separable from the other energy components.
This hypothesis
arises naturally since energy is used primarily for two purposes: trans-
portation and household operation (heating, lighting and running appliances).
Under separability, the total expenditure allocation process
for each family can be broken down into three stages:
first, the allo-
cation of expenditure between other goods and the energy aggregate;
second, the allocation of energy expenditures to gasoline and a "house-
hold operation" aggregate; and, finally, the allocation of household
operation funds to electricity, natural gas and other fuels.
Separability of gasoline allows the function (22) to be written
in the form:
T
v
kt "
where the index
4
p 1r
p 2t
p 3t
<VhT TT TT
•
refers to gasoline.
p 4t
•
\,>
•
ITT
•
\t>
Within the context of the trans-
log function (24), necessary and sufficient conditions for this structure are:
<30)
37
3
41
=
B.
=
(i = 1,
,
4
...
(31)
4)
which comprise two independent restrictions given homogeneity and sym-
metry conditions.
Within the context of the exact aggregation model, failure
to reject separability implies that each household's utility func-
tion displays separability of gasoline from the other fuels.
This
implies that a "household operation" aggregate price and quantity could
be constructed for each family.
However, the appearance of demographic
differences in the estimated equations implies that the aggregate price
of "household operations" will vary from household to household.
The
restrictions implying a common aggregate price are much stronger, requiring all demographic effects to vanish from the electricity, natural gas
and fuel oil and other equations; these restrictions are certainly in
conflict with the significance of the demographic coefficients noted
above.
We perform the test of separability, as indicated in Section
by first estimating the model subject to the restrictions (31), and
then comparing S (p) -
S
(6)
to x
2
(2)
critical values, where
the restricted objective function value.
Table
estimates constrained to obey separability.
3
S
(p)
is
contains the NL3SLS
In comparison to Table 1,
we see that virtually all demographic coefficients are identical.
For
the price coefficients, we see that those in the electricity equation
3,
38
Table
3
Pooled Estimation Results
With Separability Imposed
P
E
39
increase somewhat in absolute value, with the remaining (unconstrained)
coefficients declining in absolute value.
The test of separability is performed as follows: S (p) =
85375.61, S(9) = 85366. 44, so that S (p) - S(6) = 9.17.
The one percent
2
critical level for a x (2) is 9.21, so we fail to reject separability
at the one percent level of significance.
With the amount of data
employed, a one percent level of significance is reasonable; if anything,
it may be insufficiently stringent.
Further evidence of the acceptabil-
ity of separability is given by Tables
lated from the estimates of Table 3.
4
A-E, where elasticities calcu-
A comparison with Tables
2
A-E
indicate that the two sets of elasticities are almost identical, even
with the gasoline elasticities constrained to -1.00 under separability.
TABLE
4
40
Own Price Elasticities
A.
Varying Family Size
Electricity
Natural Gas
Fuel Oil &
Other
Gasoline
:
.,:«»» -...»»«
,«.77,-».77.2»> T,....n.i
.,',,,4., -,.2412,2
./„„„
-
-..«""
B.
Fuel Oil & Other
Gasoline
—
8
-
1 '
,98iM
,
-
21
" 69
"'•"'
-oooooo -uoooooo -..oooooo-..^
Varying Age of Head
55-64
45-54.
35-A4
25-34
15-24
Natural Gas
2
or mc
-,.3,2047 -,.32,-74 ...31,
.,.3.7,12 -1.355744 -..17,444
.,.000000
./.oooooo -..oooooo
Electricity
1 -
——
7
A
3
65 or more
-l..«7..
-,.752905 -7.7J7M4 -1.758255
-,.2,1226 -1.199952 -,.154*0,
.1 353070 -,.242272 -,.,9855,
-,.330557 -1.303042 -,.257755
...39,625 M.3,2624 -1.352047
-.oooooo -l.oooooo -t.oooooo
.,.oo„ooo -i.oooooo ...oooooo
-,.«W77 -K7U307
C.
N.E.
Varying Region
N.C.
§_
W
-1.775003
-1.75290b -1.782*53 -1.629211
-1.139611
.1.19B551 -1-123388 -1.1"l«
-1.715932
-1.352047 -1.657726 -1.80619*
-i.oooono
-l.oooooo -i.oooooo -i.oooooo
Electricity
Natural Gas
Fuel Oil & Other
Gasoline
Varying Race
Nonwhite
White
D.
E.
Varying Type of Residence
Rural
U r b an
-1.752905
Electricity
-1.702878
-1.797952
-1.752905
-1.198551
Natural Gas
-1.114202
-4.113869
-1.198551
-1.229052
-1.352047
-1.360708
-1.352047
Fuel Oil & Other
-1.000000
-1.000000
-1.000000
-1.000000
Gasoline
41
Footnotes
1.
An entirely different motivation for an exact aggregation
procedure is obtained by making assumptions on the distribution of x
for each
kt
Namely, one can assume that (2) is the correct macro model,
t.
and that x
is a sufficient statistic for the underlying distribution.
Then, cross section regressions (as in the pooling procedure) consistently
estimate the first derivatives of the macro function, i.e.
3(p
,
6).
See Stoker (1981) for the justification of this approach.
2.
There is a large literature on pooling time series and
cross section data in a linear format
and Nerlove (1966).
others.
—
the classic paper is Balestra
See also Maddala (1971) and Mundlak (1978), among
In fact, our procedure can be viewed as a full pooling estima-
tor, with missing cross section data for certain time periods.
Stoker (1978) for an exposition of this point.
See
Also it should be noted
that most discussions of pooling time series and cross-section data are
concerned with stochastic structure and not the structural model given
here.
Issues regarding fixed and random effects as well as heterosced-
asticity problems (see for example, Amemiya (1978)) can be studied within
the framework of this paper although we do not explicitly treat them.
3.
For any nonsingular error structure there always exists a
diagonalizing transformation; however, the transformations of greatest
practical interest do not Involve the distribution of x
across the popu-
lation and allow the error structure of the model to be analyzed using
either the cross section or time series data base individually.
Stoker
42
(1978) gives an example where individual errors are serieally correlated
but a simple transformation allows the correlation to be studied using
only the average data.
In the exposition we assume that the variance of the dis-
A.
conditional on the instrumental variables is constant for
turbance
both cross-section and average time series models.
is relaxed,
If this assumption
efficiency gains are possible by adjusting the weighting
matrix of equations 12 and 13.
See White (1980a, 1980b, 1982) and
Hansen (1982) for details.
5.
An exposition of the Gauss-Newton method for single equations
can be found in Draper and Smith (1966);
this method is discussed for sys-
tems of nonlinear regression equations by Malinvaud (1980).
6.
If the correction (8)
is applied x*
should replace x
o
o
here.
7.
Learner (1976)
8.
Matrix weighted averages are discussed in Chamberlain and
and Mundlak (1978), among others.
The model, data and results are discussed in much greater
detail in Jorgenson, Lau and Stoker (1981).
9.
The theory of multi-stage budgeting is reviewed in Black-
orby, Primont and Russell (1978).
10.
For these restrictions see Jorgenson and Lau (1975).
11.
In our previous notation, p
prices and x
tics at time
is the vector of component
refers to the constant and attribute-expenditure statist.
43
12.
Nonlinear three stage least squares estimates are invari-
ant to the choice of omitted equation.
13.
Expenditures on all four components are observed in each
data set except the 1960-61 CES, where electricity and natural gas are
combined.
Utilizing this data requires only a trivial modification in
the construction of the appropriate moments of M
and M
.
See Jorgenson,
Lau and Stoker (1981) for details.
14. More formally,
the model in each cross section is a multi-
variate linear regression model with respect to the coefficients 6(p
,
8).
Best linear unbiased estimation of such a model is equivalent to OLS
applied to each equation.
15. We list these instruments in the Appendix.
16.
The existence of demographic effects contradicts the existence
of a common aggregate price of energy for all households.
44
References
Amemiya, T. (1977), "The Maximum Likelihood and
Nonlinear Three-Stage
Least Squares Estimator in the General Nonlinear
Simultaneous
Equations Model," Econometrica 45, 955-968.
(1978), "A Note on a Random Coefficients Model," International
Economic Review Vol. 19, pp. 793-796.
,
Balestra, P. and M. Nerlove (1976), "Pooling Cross Section and Time
Series Data in the Estimation of a Dynamic Model: The Demand
for Natural Gas," Econometrica 34, 585-612.
Blackorby,
Boyce, R. and R.R. Russell (1978), "Estimation of Demand
C.
Systems Generated by the Gorman Polar Form; A Generalization
of the S-Branch Utility Tree," Econometrica 46, 345-364.
Blackorby,
C, Primont, D. and R.R. Russell (1975), "Budgeting, Decentralization and Aggregation," Annals of Social and Economic
Measurement 4, 49-101.
,
,
Brown, M. and D. Heien (1972), "The S-Branch Utility Tree:
A Generalization of the Linear Expenditure System," Econometrica 40,
737-747.
Chamberlain, G. and E. Learner (1976), "Matrix Weighted Averages and Posterior Bounds," Journal of the Royal Statistical Society, B
,
38,
73-84.
Deaton, A. and J. Muellbauer (1980), "An Almost Ideal Demand System,"
American Economic Review 70.
,
Draper, N.R. and H. Smith (1966), Applied Regression Analysis
New York.
,
Wiley,
Gallant, R. (1977), "Three-Stage Least-Squares Estimation for a System
of Simultaneous, Nonlinear, Implicit Equations," Journal of
Econometrics , 5, 71-88.
Gallant,
and D.W. Jorgenson (1979), "Statistical Inference for a System of Simultaneous, Nonlinear, Implicit Equations in the Context of Instrumental Variable Estimation," Journal of Econometrics 11, 275-302.
R.
,
Hansen, L.P. (1982), "Large Sample Properties of Generalized Methods of
Moments Estimators," Econometrica , forthcoming.
(1961), "The Modified Gauss-Newton Method for the Fitting
of Non-Linear Regression Functions by Least Squares," Techno-
Hartley, H.O.
metrics, 3, 269-280.
45
Jorgenson, D.W. and J. Laffont (1974), "Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances," Annals
of Economic and Social Measurement
3, 615-640.
,
Jorgenson, D.W. and L.J. Lau (1975), "The Structure of Consumer Preferences," Annals of Social and Economic Measurement 4, 49-101.
,
Jorgenson, D.W.
L.J. Lau, and T. Stoker (1981), "Modeling Energy Expenditures Through Exact Aggregation," Einal Report to the Electric
p ower Research Institue, Inc., RF-1428-1.
,
no«2\ "T*e T*\insvendonta1
,
_______ an-'
Logarithmic Model of Aggregate Consumer Behavior," in R. Basm.inn
and 0. Rhodes, eds., Advances in Econometrics Vol. 1, Greenwich,
JAI Press, pp. 97-238.
,
Kuhn, H.W. and A.W. Tucker (1951), "Nonlinear Programming," in J. Neyman,
Proceedings of the Second Berkeley Symposium on Matheed.
Berkeley, University of
mat icnT St at is t les n m\ Probnhi 1 ty
1Pre
ss
48
4
9
California
,
2
,
i
,
Lau, L.J.
"Existence Conditions for Aggregate Demand Functions,"
Econometrica , forthcoming.
Liew, C.K.
(1976), "A Two-Stage Least Squares Estimator with Inequality
Restrictions on Parameters," Review of Economics and Statistics , 58, 234-238.
(198
),
Maddala, G.S. (1971), "The Likelihood Approaches to Pooling Cross Section
and Time Series Data," Econometrica 39, 939-954.
,
Malinvaud,
E. 1980, Statistical Methods of Econometrics
Amsterdam, North-Holland.
,
3rd.
ed.,
Mundlak, Y. (1978), "On the Pooling of Time Series and Cross Section
Data," Econometrica , 46.
Roy, R.
(1943), De 1' Utilitie:
Paris; Herman.
Stoker, T.
Contribution a la Theorie des Choix
,
(1978), "The Pooling of Cross Section and Average Time Series
Data," draft, Harvard University.
(1979), Aggregation over Individuals and Demand Analysis ,
unpublished doctoral dissertation, Harvard University.
(1981), "The Use of Cross Section Data to Characterize Macro
Functions," forthcoming in Journal of the American Statistical
Association.
46
White, H.
(1980a), "Nonlinear Regression on Cross-Section Data," Econometrica, Vol. 48, pp. 721-746.
(1980b), "A Heteroscedasicity-Consistent Covariance Matrix
Estimator with a Direct Test for Heteroscedasticity ," Econometrica Vol. 48, pp. 817-838.
,
(1982), "Instrumental Variables Regression with Independent
Observations," Econometrica Vol. 50, pp. 483-500.
,
Zellner, A. and H. Theil (1962), "Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations," Econometrica
30, 54-78.
,
APPENDIX TABLE
AGGREGATE INSTRUMENTAL VARIABLES
The variables used as instruments for the aggregate time
series portion of the model are as follows:
11
Constant
12
TL
13
14
15
16
17
— effective tax rate, labor services
TCR — effective tax rate, noncompetitive imports
LH — time available for labor services
p — u.S. population, millions of individuals
PL — implicit deflator, supply of labor service
PLG — implicit deflator, government purchases of
labor
services
18
19
—
which equals governexogenous income
EL-HR-RT
ment transfers to persons (excepting social insurance)
less personal transfers to foreigners and personal nontax
payments to government
—
W(-l)
,
private national wealth, lagged one period
T
—
potential time for labor services;
of Harrod neutral change
110
LH- (1+H)
111
Total imports
112
PCR
113
PL/(1+H)
114
T
—
—
H
implicit deflator, noncompetitive imports
T
—
corrected deflator for labor services
time, set to
in 1972
—
rate
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